Matrix inequalities can be dilated in order to obtain a larger matrix inequality. This can be a useful technique to separate design variables in a BMI (bi-linear matrix inequality), as the dilation often introduces additional design variables.
A common technique of LMI dilation involves using the projection lemma in reverse, or the "reciprocal projection lemma." For instance, consider the matrix inequality
where
,
, with
This can be rewritten as
(1)
Then since
which is equivalent to
(2)
These expanded inequalities (1) and (2) are now in the form of the strict projection lemma, meaning they are equivalent to
(3)
where
and
By choosing
we can now rewrite the inequality (3) as
which is the new dilated inequality.
Some useful examples of dilated matrix inequalities are presented here.
Example 1
Consider matrices
where
and
The following matrix inequalities are equivalent:
Example 2
Consider matrices
and
where
The matrix inequality
implies the inequality
Example 3
Consider matrices
and
where
The matrix inequality
implies the inequality